## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 7, Number 4 (2016), 686-690.

### Dense Banach subalgebras of the null sequence algebra which do not satisfy a differential seminorm condition

#### Abstract

We construct dense Banach subalgebras $A$ of the null sequence algebra ${c}_{0}$ which are spectral-invariant but do not satisfy the ${D}_{1}$-condition $\Vert ab{\Vert}_{A}\le K(\Vert a{\Vert}_{\infty}\Vert b{\Vert}_{A}+\Vert a{\Vert}_{A}\Vert b{\Vert}_{\infty})$ for all $a,b\in A$. The sequences in $A$ vanish in a skewed manner with respect to an unbounded function $\sigma :\mathbb{N}\to [1,\infty )$.

#### Article information

**Source**

Ann. Funct. Anal., Volume 7, Number 4 (2016), 686-690.

**Dates**

Received: 12 May 2016

Accepted: 20 July 2016

First available in Project Euclid: 5 October 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1475685115

**Digital Object Identifier**

doi:10.1215/20088752-3661242

**Mathematical Reviews number (MathSciNet)**

MR3555760

**Zentralblatt MATH identifier**

1369.46042

**Subjects**

Primary: 46L87: Noncommutative differential geometry [See also 58B32, 58B34, 58J22]

Secondary: 46H10: Ideals and subalgebras 46J10: Banach algebras of continuous functions, function algebras [See also 46E25] 46B45: Banach sequence spaces [See also 46A45] 46K99: None of the above, but in this section

**Keywords**

$D_{1}$-subalgebra spectral invariance null sequence algebra differential structure in $C^{\star}$-algebras

#### Citation

Schweitzer, Larry B. Dense Banach subalgebras of the null sequence algebra which do not satisfy a differential seminorm condition. Ann. Funct. Anal. 7 (2016), no. 4, 686--690. doi:10.1215/20088752-3661242. https://projecteuclid.org/euclid.afa/1475685115