All Questions
Tagged with oa.operator-algebras c-star-algebras
597 questions
1
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Injective element of a commutative Banach algebra
A revision:
According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$.
Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal ...
- 356
0
votes
0
answers
134
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semifinite projection
Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$.
( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
- 101
11
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0
answers
259
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Status of the analog of the Haar measure on quantum groups
In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...
- 561
4
votes
1
answer
148
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Crossed Products by Permutation Groups
What can be said about the following crossed product $C^*$-algebra?
Let $A$ be a Kirchberg algebra with $K_0(A) = \mathbb{Q}$ and $K_1(A) = 0$. Consider the direct sum of $n$ copies of $A$, i.e. $B = ...
- 7,637
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270
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Are two $W^*$-algebras $A$ and $B$ that are Morita equivalent (in the sense of Rieffel) also Morita equivalent in the algebraic sense (as rings)?
Let $A$ and $B$ be $C^*$-algebras, I had a question about Morita equivalence for $C^*$-algebras and $W^*$-algebras, I've come across 3 concepts:
Morita equivalence for $C^*$-algebras: Equivalence of ...
- 417
7
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0
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359
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Dense ideals in C*-algebras and K-theory
Let $A$ be a nonunital C*-algebra and let $I \subset A$ be a dense, $*$-closed, 2-sided ideal. I was under the impression that there existed some "obvious" argument proving that $I$ carries all the $K$...
- 662
4
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146
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A question on extension of $Z^{*}$ algebras
A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence of ...
- 356
1
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0
answers
198
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Connected component of the identity in graded Banach algebras
I search for a noncommutative idempotent-less Banach algebra $A$ which is graded by a finite abelian group $G$ such that a nontrivial homogenous element lies in the same connected component as $1_{...
- 356
8
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0
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362
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C* algebras of free semicircular systems
It was shown by Pimsner and Voiculescu in 1982 that the reduced group $C^{*}$-algebras $C^{*}_{r}(\mathbb{F}_{n})$ and $C^{*}_{r}(\mathbb{F}_{m})$ are isomorphic if and only if $n = m$ (here, $\mathbb{...
- 629
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topology on the automorphism group of a C* algebra
Let $A$ be a $C^*$-algebra. The group $Aut(A)$ of $\ast$-automorphisms of $A$ is usually equipped either with the pointwise norm topology, i.e. the topology generated by the semi-norms $\lVert \varphi ...
- 7,637
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483
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The approximation property of group C*-algebras
Let $G$ be a discrete group. Then the group C*-algebra $C^*(G)$ is nuclear if and only if $G$ is amenable. I am wondering whether nuclearity of $C^*(G)$ can fail for a Banach-space reason, namely due ...
- 11.3k
10
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605
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Given a C-star dynamical system and a subgroup of the acting group, is the corresponding map on crossed product algebras necessarily an injection
Let $(A,\alpha, G)$ be a $C^*$-dynamical system, where $G$ is a discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes_\alpha \Gamma$ and $A\...
- 1,702
1
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1
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121
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Quick question about conjugate equivalence bimodules and inner products
Let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, link:http://...
- 417
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0
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137
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A continuous choice of invertible elements
Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
Is there a continuous map $\alpha:A^...
- 356
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1
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287
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All AI-algebras are AT-algebras
It is known that every AI-algebra (i.e. inductive limit of interval algebras) is an AT-algebra (i.e. inductive limit of circle algebras)?
This seems a little bit odd because a building block of an AT-...
- 169
2
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1
answer
323
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Ideal spanned by matrix units isomorphic to compact operators
Hello,
Assume we have $(n+1)$ isometries $S_1,...,S_{n+1}$ in the separable Hilbert space $H$ with the properties that $\sum_{i=1}^{n+1}S_iS_i^*=I, S_i^*S_j=0$ (i.e. $S_i$ are the generators of the ...
- 23
2
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2
answers
1k
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Why is this a conditional expectation into the fixed point algebra?
Let $A$ be a C*-algebra and let $\alpha$ be an action of the circle group $S_1$ on $A$ (Gauge action).
We define the following map:
$$E:A\rightarrow A;\quad E(a):=\int\alpha_t(a)\textrm{d} t.$$
My ...
- 23
5
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1
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318
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What's the link between the Toeplitz operators on H^2 and those used to define Cuntz-Pimsner algebras?
An alternate way to phrase this question might be, "How did the Toeplitz operators used in the definition of the Cuntz-Pimsner algebra come by their name?" or, "What's the relationship between the ...
- 151
3
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0
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109
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Two questions on topological and geometric structure of projections in a simple $C^{*}$ algebra
Let $A$ be a simple $C^{*}$ algebra. Assume that the space of projections has a connected component homeomorphic to the complex Grassmanian $G(k,n)$. Is it true to say that, for all $k'<k$, the ...
- 356
1
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1
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67
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Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact manifold
Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$.
This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the ...
- 356
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293
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Lifting triangles in K-theory to KL-groups
Let $X$ and $Y$ be finite simplicial complexes (or $CW$-complexes) so that $Y\subseteq X$. Let $s\colon C(X)\to C(Y)$ be the map given by restriction. In particular $K_{*}(C(X))$ and $K_{*}(C(Y))$ are ...
2
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1
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391
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when does a $C^*$-algebra have no nonzero unital quotient?
In their paper: "Addition of $C^*$-algebra extensions", G. A. Elliott and D. E. Handelman have discussed some relation between traces and equivalence of projections in $M(A)$, where $M(A)$ is the ...
- 147
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1
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243
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Unitary with full spectrum
I have a unitary element $u\in C(\mathbb{T},M_{n}(\mathbb{C}))$ such that $Spec(u)=\mathbb{T}$. Does there exist a unitary $v\in C(\mathbb{T},\mathbb{C})$ such that $Spec(uv)\subsetneqq\mathbb{T}$?
- 169
14
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0
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2k
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Schwartz kernel theorem for A-linear operators
Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...
- 7,637
5
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3
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633
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Ideal of "Compact Operators" in a W*-algebra which gives the sigma-strong-* topology.
In the case of bounded operators on a Hilbert space $\mathcal{H}$, $L(\mathcal{H})$, there are multiple descriptions of the $\sigma$-strong-* topology, namely:
1) As given by seminorms $p_{\phi},~p_{\...
- 83
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1
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1k
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Does equality of the operator norm and the cb norm for every bimodule map over a C*-subalgebra imply that the subalgebra is matricially norming?
In this post, without further mention all C*-algebras are assumed to have an identity element and subalgebras inherit the identity.
Question: Let $\mathcal{C}$ be a C*-subalgebra of $\mathcal{B}$. ...
- 7,329
4
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0
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374
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Hans Saar's thesis
I would love to have a look on some results which are claimed by some people to be in Saar's thesis:
H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C${}^\ast$-...
- 505
4
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1
answer
126
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Automorphisms of "rational" Kirchberg algebras
Let $M_{\mathbb{Q}}$ be the universal UHF-algebra and let $\mathcal{O}_{\infty}$ be the infinite Cuntz algebra. Let $A$ be a Kirchberg algebra that satisfies the UCT with $K_0(A) \cong \mathbb{Q}^n$ ...
- 7,637
10
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508
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Tensorial decomposition of $B(H)$
Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
- 101
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0
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129
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A special Lie subalgebra
Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra $$...
- 356
2
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0
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171
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tensor product of the disc algebra with itself
Let $A=\mathcal{A}(\mathbb{D})$ be the disc algebra. Is there a cross norm on $A\otimes A$, which is compatible to the Banach algebra multiplication, such that the resulting Banach algebra has a $C^{*}...
- 356
2
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1
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391
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When C(K) is closed in sigma strong topology?
Fix a compact Hausdorff space $K$ and think about $C(K)$ as a C*-algebra acting on a Hilbert space $H$. Suppose that $C(K)$ is closed in $\mathcal{B}(H)$ in:
$\sigma$-strong
$\sigma$-strong*
...
- 11.3k
3
votes
2
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429
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Kernel projections in the universal representation.
Let $A \subseteq \mathcal B(\mathcal H)$ be a unital C*-algebra in its universal representation. The GNS representation $\pi_\mu\colon A \rightarrow \mathcal B(\mathcal H_\mu)$ with base state $\mu$ ...
- 1,199
7
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0
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189
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Replacing commutative C*-algebras by simple ones
I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$.
I am particularly ...
- 7,637
5
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1
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410
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Is the unitary group of $l^2(A)$ with the strict topology contractible?
Let $A$ be a $C^*$-algebra with countable approximate unit. Let $\mathbb{K}$ denote the compact operators on a separable Hilbert space. Mingo and later Cuntz and Higson have shown that the unitary ...
- 7,637
3
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0
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170
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Closure of pseudodifferential operators of order 0 on compact manifolds
Let $M$ be a compact manifold. If we have a pseudodifferential operator $P$ of order $0$ on $M$, then $P$ is pseudolocal, i.e., every commutator $[f,P]$ with a continuous function $f \in C(M)$ is a ...
- 2,998
28
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2k
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Finite-dimensional subalgebras of $C^\star$-algebras
Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
- 25.5k
7
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0
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573
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References for "folklore" on strong amenability of (group) C*-algebras?
[Apologies in advance for the prolixity - but I was unsure how much of the story is familiar.]
$\newcommand{\ptp}{\widehat{\otimes}}
\newcommand{\co}{\operatorname{co}}
\newcommand{\Cst}{\operatorname{...
- 25.8k
2
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1
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653
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Strict positivity in dense subalgebras of $C^{*}$-algebras
Let $A$ be a $C^{*}$-algebra, represented on a Hilbert space $H$, and $D$ a selfadjoint unbounded operator on $H$ (note that we do not impose that $D$ have compact resolvent). Let
$\mathcal{A}:=${$a\...
- 213
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1
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232
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simultaneously Approximated by self-adjoint elements.
We know that a $C^*$-algebra $A$ has real rank zero iff every self-adjoint element of $A$ can be approximated in norm by self-adjoint element with finite spectra. My question is:
If we have two self-...
- 147
8
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1
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460
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Finite dimensionality of certain $C^{\star}$-algebras
In the discussion about the question Finite-dimensional subalgebras of $C^{\star}$-algebras the following separate question came up:
Let $H$ be a Hilbert space and $a_1, \dots, a_n \in B(H)$ be self-...
- 25.5k
5
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2
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862
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Hilbert $C^*$-modules and approximate units
Hi,
Given a $\sigma$-unital $C^*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon_i\}, i\in I$ in $A$ such that each $\epsilon_i$ is of the ...
- 93
4
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0
answers
282
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Extensions of completely positive mappings
I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-...
- 619
4
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1
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734
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Approximate unit for the algebra C*(h) consisting of projectors
Let E be a Hilbert C*-module over some C*-algebra and let $h \in K(E)$. Due to B. Blackadar's, "K-Theory for Operator algebras" Thm. 17.11.4 for a separable C*-algebra $A$, represented by ...
- 613
15
votes
0
answers
790
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Must we close weakly to apply the spectral theorem?
Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate.
The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...
- 7,329
1
vote
0
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212
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Grading on Multiplier Algebras
A graded C*-algebra A is inner-graded if there exists a self-adjoint unitary $\varepsilon$ in the multiplier algebra M(A) of A which implements the grading automorphism $\alpha$ on A: $\alpha(a)=\...
- 1,702
2
votes
0
answers
200
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Fredholmness and invertibility in a C* algebra generated convolution-type operators
Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
- 21