Questions tagged [c-star-algebras]
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].
854
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Takesaki's proof of the Kaplansky density theorem
Consider the following fragment from Takesaki's book "Theory of operator algebra I":
Why is the boxed sentence true? It looks like they replace $A$ by its strong$^*$-closure. Is this ...
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0
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136
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Representation of quantum groups
Let $\mathbb{G}=(A,\Delta_A)$ be a C*-quantum group and $\mathbb{H}=(B,\Delta_B)$ be a closed quantum subgroup of $\mathbb{G}$. We say that $\mathbb{H}$ is a closed quantum subgroup of $\mathbb{G}$ if ...
- 121
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Non-commutative harmonic analysis on the discrete Heisenberg group
Question: Is there a linear map $\mathcal F$ from the Hilbert space of $\ell^2$ functions on the discrete Heisenberg group to some Hilbert space of functions $ L^2(\bigcup \{\Omega_n\}) $, such that:...
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1
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Convergent bounded net of positive operators converges to a positive operator
Let $A$ be a $C^*$-algebra. Endow $A$ with the strict topology for which a net $\{a_i\}_{i \in I}$ converges to $a \in A$ if $$\|a_i b-ab\| + \|ba_i-ba\| \to 0$$
for all $b \in A$. Is it true that if $...
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A completely positive equivariant map $\varphi: A \to B$ induces a map on the full crossed products
Let $G$ be a discrete group. Let $(A,\alpha)$ and $(B,\beta)$ be $G$-$C^*$-algebras and $\varphi: A \to B$ be $G$-equivariant and completely positive. All crossed products in this post are full (= ...
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2
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0
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A foliation with prescribed graph of foliation
**I have already asked this question on MSE https://math.stackexchange.com/questions/4272279/1-dimensional-foliation-of-surfaces-with-prescribed-graph-of-foliation **
Definition of the graph of a ...
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0
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161
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Stinespring's theorem: can we choose the dilation to be an isometry?
Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
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2
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Can nuclearity be determined by tensoring with a single C*-algebra?
A C*-algebra is nuclear if the algebraic tensor product $A\odot B$ ($B$ is any other C*-algebra) admits a unique C*-norm. This definition requires testing the condition for nuclearity with `all' C*-...
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Kadison-Singer problem in exotic Hilbert spaces
The Kadison-Singer problem is considered in relation to the separable Hilbert space:
KS: Does every pure state on the diagonal (atomic) masa of $B(\ell_2)$ has a unique extension to $B(\ell_2)$?
...
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Action of a group $G$ induces a coaction on $C_0(G)$
In this question, I follow the book "An invitation to quantum groups and duality" by Timmerman, p259.
Let $G$ be a locally compact group and $C$ be a $C^*$-algebra. Assume an action
$$\alpha:...
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3
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1
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Woronowicz characters are multiplicative
I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset.
Let $G$ be a $C^*$-algebraic compact quantum group with function algebra $(C(G), \Delta)...
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3
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1
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Norm antipode on a Kac-type compact quantum group
Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group. Consider the associated dense Hopf$^*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ ...
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2
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Is the injective envelope functorial?
Let $A$ and $B$ be unital $C^*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^*$-...
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Can the injective envelope ever be injective for $*$-homomorphisms?
The answers to the question "Is the injective envelope functorial" resoundingly remind us that the injective envelope of a C$^*$-algebra really belongs in the category of completely positive ...
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0
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Tensor product of representations on a compact quantum group
Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$.
Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
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Subrepresentations of C*-algebraic compact quantum groups
Let $\mathbb{G}$ be a compact quantum group with function algebra $(C(\mathbb{G}), \Delta)$ (in the sense of Woronowicz). Let $X \in M(B_0(H) \otimes C(\mathbb{G}))$ be a (possibly infinite-...
- 4,011
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1
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If a completely positive unital map admits a completely positive unital left inverse, it is a complete isometry
Let $T$ be an injective operator system and $U$ be an arbitrary operator system. Let $\varphi: T \to U$ be a unital completely positive map and $\psi: U \to T$ be a unital completely positive map with ...
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$(\iota \otimes f)(X) = 0$ for all $f \in B^*$ implies $X=0$
Let $A$ and $B$ be $C^*$-algebras. Given $f \in B^*$, we can form the right slice map
$$\iota \otimes f: A \otimes B \to A: a \otimes b \mapsto af(b)$$
which extends uniquely to a bounded linear map
$$...
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0
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C^*-algebra theory with all the Koszul signs
I was wondering if someone knows of a reference in which $\mathbb{Z}_2$-graded $C^*$-algebra theory is developed using the sign convention $(ab)^* = (-1)^{|a||b|}b^* a^*$. I would be most enthusiastic ...
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Stable isomorphism of group C$^*$-algebras
For a discrete group $G$, let $C^*_r(G)$ be its reduced group C$^*$-algebra.
Question: Do there exist discrete, torsion-free non-isomorphic groups $G,H$ such that $C^*_r(G)$ and $C^*_r(H)$ are stably ...
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1
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$C\lVert\sum_i a_{ii}\rVert \ge \lVert(a_{ij})\rVert$ for matrices with entries in a $C^*$-algebra
Let $A$ be a $C^*$-algebra and $(a_{ij}) \in M_n(A)$ be a positive matrix. Does there exist a constant $C \ge 0$ (not depending on the $a_{ij}$) such that
$$\lVert(a_{ij})\rVert \le C \Bigl\lVert\...
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1
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Example of a ternary $C^{\ast}$-ring which is not an operator space
A ternary $C^{\ast}$-ring is a complex Banach space $X$, equipped with a ternary product $[\cdot,\cdot,\cdot]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle variable. ...
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Two densely defined traces on a $C^*$-algebra coinciding on a dense subalgebra are equal
Let $t_1$ and $t_2$ be lower semicontinuous semifinite densely defined traces on a $C^*$-algebra $A$. Let us denote by $\mathcal{R}_1$ and $\mathcal{R}_2$ their ideals of definition, i.e. $\mathcal{R}...
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1
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Is $N_\phi = \{x \in E: \phi(\langle x,x\rangle)=0\}$ a Hilbert submodule of $E$?
Let $E$ be a (right) Hilbert module over the $C^*$-algebra $B$. Let $\phi$ be a state on the $C^*$-algebra $B$. Then consider
$$N_\phi:= \{x \in E: \phi(\langle x,x\rangle)=0\}.$$
I want to show that $...
- 2,346
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1
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Monotone approximation of elements in AF-algebras
Suppose that we are given an AF-algebra $A$ and a sequence of finite-dimensional subalgebras $\mathbb{C}=A_0\subset A_1\subset A_2\subset\ldots$ such that $A=\overline{\bigcup\limits_{n\geq 0}A_n}$. ...
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2
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Fredholm $C^*$-algebras
Let $H$ be a Hilbert space. A vector subspace $W\subset B(H)$ is called a Fredholm subspace if there is an upper bound for the absolute value of Fredholm index of all Fredholm operators $T$ in $W$.
...
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1
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Gelfand-Naimark and Peter-Weyl for the unitary group
Consider the compact Lie groups $U(l)$ (the unitary group) and $U(1) \times SU(l)$ for some natural number $l$. Both the groups have the same Lie algebra $\frak{gl}_l$. Which means that they both have ...
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Do extensions of pure states separate points?
Let $B$ be a unital C*-algebra and let $A⊆B$ be a closed *-subalgebra containing the unit of $B$. I am mostly
interested in the case that $A$ is abelian but, for the strict purpose of stating my ...
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1
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What are the maximal ideals of $C_0 (X),$ where $X$ is a locally compact Hausdorff space?
Crossposted from MSE
How do the maximal ideals of $C_0(X)$ look like where $X$ is a locally compact Hausdorff space?
I know that if $X$ is a compact Hausdorff space then the maximal ideals of $C(X)$ ...
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1
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finitely generated C*-algebra as $C(X)$
In the question ($C(X)$ as finitely generated $C^*$-algebra), the answer show that spectrum of an abelian unital finitely generated C*-algebra is homeomorphic to compact subset of $\mathbb{C}^{n}$. I ...
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1
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Trying to recognise a $C^*$-algebra
Let $H$ and $K$ be infinite dimensional (separable) Hilbert spaces and $X=B(H,K)$ denote the space of bounded linear operators. For $T_1, T_2$ in $X$, we define $D_{T_1,T_2}:X \to X$ as $D_{T_1,T_2}(T)...
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Trying to understand morphisms in category of ternary $C^*$-rings
Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
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Hermitian vector bundles and Hilbert $C^*$-modules
Let $X$ be a compact Hausdorff space and $C(X)$ its algebra of continuous complex valued functions. The Gelfand-Naimark theorem tells us that we have a duality between commutative $C^*$-algebras and ...
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Is it true that $\varinjlim A_n^{**}=(\varinjlim A_n)^{**}$?
Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras. It's well known double dual of $C^*$-algebra is again a $C^*$ algebra.
Is it true that $\varinjlim A_n^{**}=(\varinjlim A_n)^{**}$
Can ...
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Drinfeld center of a tensor category
Firstly, apologies for the imprecise language, I'm a physicist trying to understand anyonic excitations from the lens of category theory.
If I have a category (say $\operatorname{Rep}(\mathbb{Z}_2)$) ...
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$\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$
Let $A$ be a $C^*$-algebra, $E$ be a (right) Hilbert $A$-module and $t \in \mathcal{L}_A(E)$ be an adjointable operator satisfying $t=t^*$. Is it true that
$$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
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Subalgebras of $B(H)$ consisting of all operators leaving a given finite dimensional space invariant
Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$.
Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$
So $A$ is a Banach algebra.
Can we equip $A$ ...
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2
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768
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Amenable action intuition
Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
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1
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155
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Abelian twisted reduced group C*-algebra
Let $G$ be an abelian discrete group. Then is $C_r^*(G, \sigma)$ abelian?
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Direct sum of multiplier algebras
Consider a collection of $C^*$-algebras $\{A_i\}_{i \in I}$. We can form the direct sum $$\bigoplus_{i \in I}^{c_0} A_i:= \left\{(a_i)_{i \in I} \in \prod_{i\in I} A_i: \lim_{i \in I} \|a_i\| = 0\...
- 18.5k
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0
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Quasidiagonal C*-algebras
Let $A$ be a nuclear $C^*$-algebra satisfying UCT condition. Then under what assumptions $A$ is quasidiagonal?
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Complemented submodules of a Hilbert C*-module
Let $A$ be a $C^*$-algebra and $E$ be a (right) Hilbert $C^*$-module over $A$. Assume $F$ is a closed submodule of $E$ such that $F^\perp := \{x \in E: \langle x, F\rangle=0\}$ is orthogonally ...
- 42.2k
0
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1
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Why is $q(f,g) = (f-g,0)$ not adjointable?
Let $A= C([0,1])$ and $J= \{f \in A: f(0) = 0\}$. Consider the Hilbert $C^*$-module
$E:= A \oplus J$ (with the obvious right $A$-action and inner product). I want to prove that
$$q: E \to E: (f,g) \...
- 18.5k
2
votes
1
answer
227
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Primitive ideals of minimal tensor product
Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product.
Is there any classification of primitive ideals of $A \otimes B$? (I'm mainly interested in the ...
- 2,346
2
votes
0
answers
189
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Reference request: definiitions of exact C* algebra and group C* algebra
I am writing my Ph.D. thesis and I would like to cite the specific papers where the concept of exact $C^*$ algebra and group $C^*$ algebra was defined.
In the book of Brown and Ozawa "$C^*$-...
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1
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336
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A $*$-homomorphism $C(X) \to C(Y)$ gives a continuous map $Y \to X$
Given a $C^*$-algebra $A$, we write $\Omega(A)$ for its space of characters, i.e. its non-zero algebra homomorphisms $A \to \mathbb{C}$. If $X$ is a compact Hausdorff space, it is well-known that
$$X \...
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3
votes
0
answers
182
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$C^*$-algebras over an extension of $\mathbb{Q}_p$?
I'm wondering to what extent it might be possible for the theory of $C^*$-algebras to be translated into the $p$-adic context i.e. to define 'p-adic $C^*$-algebras' over some extension of $\mathbb{Q}...
3
votes
1
answer
394
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Strict topology on the multiplier algebra
Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by
$$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
- 2,346
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votes
2
answers
179
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Unconditional Convergence of Positive Terms in a $C*$-algebra
I am reading the paper Frames and Outer Frames for Hilbert $C^*$-modules by L.J. Arambasic and D. Bakic. They have mentioned in passing, the following:
"...Since in each $C^*$-algebra, a ...
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votes
1
answer
162
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Regarding socle of a C* algebra
I wanted to know if the socle of a complex C*-algebra is essential?
Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...
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