scholarly journals DIAGONALS IN TENSOR PRODUCTS OF OPERATOR ALGEBRAS

2002 ◽  
Vol 45 (3) ◽  
pp. 647-652 ◽  
Author(s):  
Vern I. Paulsen ◽  
Roger R. Smith
Keyword(s):  
Tensor Product ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Operator Algebras ◽  
Direct Proof ◽  
Secondary 46L05 ◽  
C Algebra ◽  
Finite Dimensional ◽  
Hochschild Cohomology Group ◽  
Primary 46L06

AbstractIn this paper we give a short, direct proof, using only properties of the Haagerup tensor product, that if an operator algebra $A$ possesses a diagonal in the Haagerup tensor product of $A$ with itself, then $A$ must be isomorphic to a finite-dimensional $C^*$-algebra. Consequently, for operator algebras, the first Hochschild cohomology group $H^1(A,X)=0$ for every bounded, Banach $A$-bimodule $X$, if and only if $A$ is isomorphic to a finite-dimensional $C^*$-algebra.AMS 2000 Mathematics subject classification: Primary 46L06. Secondary 46L05

scholarly journals THE FIRST COHOMOLOGY GROUP OF THE TRIVIAL EXTENSION OF A MONOMIAL ALGEBRA

2004 ◽  
Vol 03 (02) ◽  
pp. 143-159 ◽  
Author(s):  
CLAUDE CIBILS ◽  
MARÍA JULIA REDONDO ◽  
MANUEL SAORÍN
Keyword(s):  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Trivial Extension ◽  
Monomial Algebra ◽  
Finite Dimensional ◽  
Hochschild Cohomology Group ◽  
First Cohomology Group

Given a finite-dimensional monomial algebra A, we consider the trivial extension TA and provide formulae, depending on the characteristic of the field, for the dimensions of the summands HH1(A) and Alt (DA) of the first Hochschild cohomology group HH1(TA). From these a formula for the dimension of HH1(TA) can be derived.

scholarly journals THE LIE ALGEBRA STRUCTURE ON THE FIRST HOCHSCHILD COHOMOLOGY GROUP OF A MONOMIAL ALGEBRA

2006 ◽  
Vol 05 (03) ◽  
pp. 245-270 ◽  
Author(s):  
CLAUDIA STRAMETZ
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Characteristic Zero ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Algebra Structure ◽  
Monomial Algebra ◽  
Finite Dimensional ◽  
Outer Automorphisms ◽  
Hochschild Cohomology Group

We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra Λ, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of Λ in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given.

scholarly journals A lower bound on the homological bidimension of a non-unital C*-algebra

1998 ◽  
Vol 40 (3) ◽  
pp. 435-444
Author(s):  
Olaf Ermert
Keyword(s):  
Cohomology Group ◽  
Jacobson Radical ◽  
Hochschild Cohomology ◽  
Equivalence Classes ◽  
Basic Question ◽  
Homological Dimension ◽  
C Algebra ◽  
Wedderburn Decomposition ◽  
Hochschild Cohomology Group ◽  
Closed Subalgebra

Let A be a C*-algebra. For each Banach A-bimodule X, the second continuous Hochschild cohomology group H2(A, X) of A with coefficients in X is defined (see [6]); there is a natural correspondence between the elements of this group and equivalence classes of singular, admissible extensions of A by X. Specifically this means that H2(A, X) ≠ {0} for some X if and only if there exists a Banach algebra B with Jacobson radical R such that R2 = {0}, R is complemented as a Banach space, and B/R ≅ A, but B has no strong Wedderburn decomposition; i.e., there is no closed subalgebra C of B such that B ≅ C © R. In turn this is equivalent to db A ≥ 2, where db A is the homological bidimension of A; i.e., the homological dimension of A#, the unitization of A, as an,A-bimodule [6, III. 5.15]. This paper is concerned with the following basic question, which was posed in [7].

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

scholarly journals Algebra endomorphisms and derivations of some localized down-up algebras

2014 ◽  
Vol 14 (03) ◽  
pp. 1550034 ◽  
Author(s):  
Xin Tang
Keyword(s):  
Automorphism Group ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Root Of Unity ◽  
Hochschild Cohomology Group ◽  
Algebra Endomorphism

We study algebra endomorphisms and derivations of some localized down-up algebras A𝕊(r + s, -rs). First, we determine all the algebra endomorphisms of A𝕊(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A𝕊(r + s, -rs) is an algebra automorphism if rmsn = 1 implies m = n = 0. When r = s-1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A𝕊(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A𝕊(r + s, -rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r + s, -rs) is an algebra automorphism in either case. Second, we determine all the derivations of A𝕊(r + s, -rs) and calculate its first degree Hochschild cohomology group.

On Simple Connectedness of Minimal Representation-Infinite Algebras

2015 ◽  
Vol 22 (04) ◽  
pp. 639-654
Author(s):  
Hailou Yao ◽  
Guoqiang Han
Keyword(s):  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Closed Field ◽  
Minimal Representation ◽  
Algebraically Closed Field ◽  
Hochschild Cohomology Group ◽  
Simple Connectedness ◽  
Simply Connected ◽  
Equivalent Conditions ◽  
Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

scholarly journals Oriented Algebras and the Hochschild Cohomology Group

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 237
Author(s):  
Ali Koam
Keyword(s):  
Associative Algebra ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Chain Complexes ◽  
Hochschild Cohomology Group ◽  
Equivariant Version

Koam and Pirashivili developed the equivariant version of Hochschild cohomology by mixing the standard chain complexes computing group with associative algebra cohomologies to obtain the bicomplex C ˜ G * ( A , X ). In this paper, we form a new bicomplex F ˘ G * ( A , X ) by deleting the first column and the first row and reindexing. We show that H ˘ G 1 ( A , X ) classifies the singular extensions of oriented algebras.

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