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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: [tag:banach-algebras], [tag:von-neumann-algebras], [tag:operator-algebras], [tag:spectral-theory].

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For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is: For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \... 1answer 94 views ### Linear independency and compactness of the set of pure states of a C^*-algebra Let \mathcal{A} be a noncommutative C^*-algebra and PS(\mathcal{A}) be the set of its pure states. Question 1. Is PS(\mathcal{A}) linearly independent (as vectors over \mathbb{R})? (If \... 2answers 293 views ### Is a C*-algebra with an isomorphic predual a von Neumann algebra? It is well-known that a C*-algebra A is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space X such that A is isometrically ... 1answer 250 views ### Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum? Let A = \mathcal{C}(X) be a commutative (unital) C*-Algebra. Let Spec(A) denote its Gelfand spectrum$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$Now ... 1answer 349 views ### Is \mathcal{K}(H) injective \mathcal{B}(H)-module? Does anyone know if the right Banach \mathcal{B}(H)-module \mathcal{K}(H) is injective? The same question for \ell_\infty-module c_0. Both these modules are not dual, so standard arguments ... 5answers 3k views ### Reference: Learning noncommutative geometry and C^* algebras I am starting to study noncommutative geometry and C^* algebras so my question is Does anyone know a good reference on this subject? I would like a basic book with intuitions for definitions and ... 1answer 136 views ### The C*-envelope of the algebra of continuous functions on a compact topological space is commutative In my research in operator theory, specifically in C* algebras and enveloping, I came across this strange footnote in a text (locally published in non English where I study) which states the following:... 1answer 75 views ### commutativity of a diagram in cohomology of C^*-algebras The setting is the same as in my last question commutative diagram with K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R}) (for C^*-algebras) : Let A be in the bootstrap category (=N in the other ... 1answer 173 views ### What does it tell us, if we know a unital C*-algebra has approximately inner (half-)flip? This is a somewhat vague question, but I think it is not too open-ended and should admit well-circumscribed answers by specialists in operator algebras.\newcommand{\Cst}{{\rm C}^*} It arises from ... 1answer 99 views ### commutative diagram with K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R}) (for C^*-algebras) I have a question about a proof in Rosenberg and Schochet's paper "the Künneth theorem and the Universal Coefficient Theorem for Kasparov's generalized K-functor", proposition 2.6. First of all, the ... 0answers 42 views ### bootstrap class (C^*-algebras): comparison of two definitions I want to clarify the relationship between two (at first sight) different definitions of the bootstrap class for C^*-algebras, in order to understand which C^*-algebras satisfy the universal ... 0answers 362 views ### Amenability of an “almost Hamiltonian” group Here is another interesting question that I can't answer on my own. Let G be a countable, discrete group such that for any subgroup H of G and any element s of G we have [H : sHt] is ... 2answers 120 views ### Multiplier algebra of A \otimes \mathcal{K} If A is unital C^*-algebra, is it true that the multiplier algebra of A \otimes \mathcal{K}  is  A \otimes \mathcal{B}(\mathcal{H})? Where \mathcal{K} is C^*-algebra of compact operators ... 5answers 764 views ### If two projections are close, then they are unitarily equivalent Given two projections p,q\in B(H), it is well-known that if \|p-q\|<1, then there exists a unitary u\in B(H) with q=upu^*. The proof that immediately occurs to me uses comparison of ... 1answer 97 views ### busby invariant of extensions of C^*-algebras I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras". Let 0\to B\to E\to A\to 0 be a short exact ... 2answers 449 views ### Kazhdan's property (T) vs. residual finiteness I have asked this question already on mathstackexchange but got no answer (see http://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ... 2answers 234 views ### id:A\to A^{op} is completely positive iff A is abelian Let A be a C^*-algebra and A^{op} it's opposite C^*-algebra. Let id:A\to A^{op} be the identity map. id is positive. The claim is: id is completely positive iff A is abelian. I need ... 1answer 155 views ### Uniform Roe algebras and exact groups Let \Gamma be a discrete group. Q: If l^\infty(\Gamma)\rtimes \Gamma=l^\infty(\Gamma)\rtimes_r \Gamma canonically, can we conclude that \Gamma is an exact group? The converse implication is ... 0answers 61 views ### A point on the absolute value of a bounded linear functional. Let A_0 be a C*-subalgebra in a C*-algebra A. Let \phi_0 be a bounded linear functional on A_0 and assume \phi is an extension of \phi_0 on A. I mean \phi\in A^* with \phi_{|_{A_0}}=\... 0answers 96 views ### An estimate for the maximal C^*-norm in the group algebra of a free group Let F\twoheadrightarrow G be an epimorphism of groups, F being finitely generated and free. Let H be its kernel. Consider a lifting i:G\hookrightarrow F of the epimorphism. Every element of ... 1answer 119 views ### is the maximal tensor product of compact operators an essential ideal? I'm searching for a counterexample for C^*-algebras A and B and essential ideals (I assume an ideal to be closed and only two-sided ideals) I\subseteq A, J\subseteq B , such that the ideal ... 1answer 125 views ### is a linear map on an operator system into a C^*-algebra (+ extra conditions) positive? First of all, sorry for my bad english. I tried to find out whether the following statement is true or not: Let X be a operator system, B a C^*-algebra and f:X\to B linear such that f(1)\ge ... 4answers 425 views ### unitization-process of unital- and non-unital C^*-algebras I have a small question about unitization of (unital) C^*-algebras. I first asked on math.stackexchange because it is basic theory, but I still have no suitable answer, the link http://math.... 0answers 92 views ### Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff? Recall that a locally compact group G is said to be an [FC]^- group, if each conjugacy class in G has a compact closure; an [SIN] group, if each neighborhood of the identity includes a ... 0answers 139 views ### The functional equation T(x\otimes y)=T(x)\otimes T(y) on certain C^{*} algebras Is there a name for the following property of a C^{*} algebra A?$$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$Example of this situation is A=C(X) where X is the ... 2answers 1k views ### The functoriality of group C* algebra structure Let G and H be discrete groups and f:G \rightarrow H be any homomorphism of these groups. I have three questions about it: 1) How to prove the functoriality of the construction of universal C^*... 0answers 81 views ### Lower bound for the C^*-unitisation norm? Let A be a C^*-algebra without unit. Its unitisation A^+ carries the C^*-norm$$\|(a,\lambda)\|_{A^+} = \sup \{\|ax+\lambda x\|_A \;\big|\; \|x\|_A = 1 \},$$which is the operator norm of ... 1answer 219 views ### On the second dual of C[0,1] I have two questions on the second dual of C[0,1]: R. D. Mauldin ([1]) proved that: For a given bounded linear functional T: C[0,1]^*\to \mathbb{C} there is a bounded function \psi defined on ... 1answer 127 views ### Representations of Calkin algebra Let H be a separable Hilbert space and consider the Calkin algebra C(H)=\frac{B(H)}{K(H)}. Q) True or false: Any representation of C(H) is a direct sum of irreducible representations. 1answer 122 views ### A point-wise separation Hahn-Banach theorem in C*-algebras Let H be a Hilbert space. We denote K(H) by the space of compact operators on H which is a two sided ideal in B(H). Let E be a norm closed convex subset of positive operators in K(H) ... 1answer 209 views ### When does a C^*-algebra have no nonzero projection? Let A be a C^*-algebra and \hat{A} its spectrum of A,the set of classes of non-zero irreducible representation of A endowed with hull-kernel topology. suppose \hat{A} is a non-compact ... 0answers 60 views ### Finding the infimum of the range of a certain non-negative function associated to a  C^{*} -algebra Let  A  be a non-trivial  C^{*} -algebra and  n \in \mathbb{N} . Setting  \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} , we can define a function  f: \... 1answer 196 views ### Is this a characterization of commutative C^{*} algebras? Assume that A is a C^{*} algebra with self adjoint elements A_{sa}. Assume that for all a,b\in A we have$$ab\in A_{sa} \iff ba \in A_{sa}$$Is A necessarily a commutative ... 1answer 263 views ### A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity In Wegge-Olsen’s book K-Theory and C ^{*} -Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of ... 1answer 122 views ### C(X) as finitely generated C^*-algebra Let X be a Hausdorff space. Suppose that C(X) (or C_0(X)) is a finitely generated C^*-algebra. What we can say about X ? For example can we characterize its inductive dimension, axioms of ... 1answer 189 views ### How “nondegenerate” are amalgamated free products of C*-algebras? In the following, I assume all algebras are unital. Let A and B be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra C. Let A *_C B denote the amalgamated free ... 1answer 281 views ### Weakly amenability and exactness for discrete groups A countable discrete group \Gamma is said to be weakly amenable with Cowling-Haagerup constant 1 if there exists a sequence of finitely supported functions (\phi_n) on \Gamma such that \phi_n\... 0answers 167 views ### A functor on the category of rings, algebras or compact Hausdorff topological space Assume that R is a unital ring or a complex or real (Banach or C^{*}) algebra. We define a relation M on R as follows:$$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...
Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...